Ergodic dynamical systems over the cartesian power of the ring of p-adic integers

For any 1-lipschitz ergodic map F: ? _p ^k ? ? _p ^k , k >1 ∈ ?, there are 1-lipschitz ergodic map G: ? _p ? ? _p and two bijections H _k , T _{k, P} that $G = H_k \circ T_{k,P} \circ F \circ H_k^{ - 1} andF = H_k^{ - 1} \circ T_{k,P - 1} \circ G \circ H_k $ .     

Relation between the functors Hom and Tor over the ring of p-adic integers

Translated from Matematicheskie Zametki, Vol. 55, No. 1, pp. 80–87, May, 1994.     

Goppa codes over the p-adic integers and integers modulo pe

Goppa codes were defined by Valery D. Goppa in 1970. In 1978, Robert J. McEliece used this family of error-correcting codes in his cryptosystem, which has gained popularity in the last decade due to its resistance to attacks from quantum computers. In this paper, we present Goppa codes over the p-adic integers and integers modulo $p^e$. This allows the creation of chains of Goppa codes over different rings. We show some of their properties, such as parity-check matrices and minimum distance, and suggest their cryptographic application, following McEliece's scheme.     

Value-sets of polynomials at p-adic integers

In this paper we are concerned with the classification of the subsets A of ${\mathbb{Z}_p}$ which occur as images ${f(\mathbb{Z}_p^r)}$ of polynomial functions ${f:\mathbb{Z}_p^r\to \mathbb{Z}_p}$ , limiting ourselves to compact-open subsets (i.e. finite unions of open balls). We shall prove three main results: (i) Every compact-open ${A\subset \mathbb{Z}_p}$ is of the shape ${A=f(\mathbb{Z}_p^r)}$ for suitable r and ${f\in\mathbb{Z}_p[X_1,\ldots ,X_r]}$ . (ii) For each r ₀ there is a compact-open A such that in (i) we cannot take r < r ₀. (iii) For any compact-open set ${A\subset \mathbb{Z}_p}$ there exists a polynomial ${f\in\mathbb{Q}_p[X]}$ such that ${f(\mathbb{Z}_p)=A}$ . We shall also discuss in more detail which sets A can be represented as ${f(\mathbb{Z}_p)}$ for a polynomial ${f\in\mathbb{Z}_p[X]}$ in a single variable.     

Haar system on the product of groups of p-adic integers

Mathematical Notes - We present an algorithm for constructing dilation operators on the product of groups of p-adic integers and construct a system of Haar functions which is obtained from a single...     

Approximation lattices of p-adic numbers

Approximation lattices occur in a natural way in the study of rational approximations to p-adic numbers. Periodicity of a sequence of approximation lattices is shown to occur for rational and quadratic p-adic numbers, and for those only, thus establishing a p-adic analogue of Lagrange's theorem on periodic continued fractions. Using approximation lattices we derive upper and lower bounds for the best approximations to a p-adic number, thus establishing the p-adic analogue of a theorem of Hurwitz.     

Some metric properties of p-adic numbers

Siberian Mathematical Journal -     

p-adic zeta functions and Bernoulli numbers

One gives a new proof to the Leopoldt-Kubota-Iwasawa theorem regarding the possibility of the p-adic interpolation of the values of the Riemann zeta-function and of the Dirichlet L-functions at negative integral points. To this end, for each root ? ≠ 1 of unity one introduces and one investigates the numbers C_n(?) which arise in the expansion $$\frac{{\varepsilon - 1}}{{\varepsilon e^z - 1}} = \sum\limits_{n = 0}^\infty {\frac{{C_n (\varepsilon )}}{{n!}}Z^n }$$ One proves a generalization of the Kummer congruences for the Bernoulli numbers.     

Approximation of p-adic numbers by algebraic numbers of bounded degree

The approximation of p-adic numbers by algebraic numbers of bounded degree is studied. Results similar to those obtained by Wirsing and by Davenport and Schmidt in the real case are proved in the p-adic case. Unlike the real case the expected best exponent is not obtained when approximating by quadratic irrationals.     

Describing the real numbers in terms of integers

The real numbers are described in terms of near-endomorphisms of the additive group of integers.Received: 9 January 2005     

Riesz products on the ring of dyadic integers

A class of Riesz products on the ring of dyadic integers is introduced. Using almost everywhere convergence of certain series, Hausdorff dimension of these Riesz products is determined. Other properties, such as mutually absolute continuity, quasi-invariance and quasi-Bernoulli property, are also discussed.